Q.If the tangent at $(3,-4)$ to the circle $x^2+y^2=25$ touches the circle $x^2+y^2+8x-4y+c=0$, find $c$.

For a line of unknown position, written generally as $y=mx+c$, tangency to the circle $x^2+y^2=a^2$ is decided by comparing it against the tangent-equation pattern at an as-yet-unknown point of contact and eliminating that point's coordinates; the result is a clean condition purely in terms of $m$, $c$, and $a$:

Equivalently, $c=\pm\sqrt{a^2m^2+a^2}$ — meaning that for any slope $m$, there are always exactly two tangent lines to a given circle with that slope, one on each side of it. This condition is the standard tool whenever a tangent's slope is known (or constrained — parallel to a given line, perpendicular to one, or a stated numeric slope) but its exact position is not: substitute the slope, solve for $c$, and the tangent line(s) follow directly. An equivalent and often faster check for the same question is purely …